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AIEEE Mathematics Quick Review
COMPLEX NUMBERS AND DEMOIVRES THEOREM
1
...

2
...
Product of nth root of unity (–1)n–1
4
...
1 + ω + ω2 = 0, ω3 = 1,
−1 + 3i 2 − 1− 3i
ω=
,ω =
2
2

of z represents Ellipse and if k < z1 − z2

x > 0, y > 0

S

x+a
x−a
+i
, a − ib
2
2

nπ
15
...
If x=Cosθ+iSinθ then

K

x+a
x −a
−i
where x =
2
2

z1 + z 2 ≥ z 1 − z 2 ;

z1 − z 2 ≥ z 1 − z 2

S

If three complex numbers
Z1, Z2, Z3 are collinear then
 z1

 z2
z
 3

z1 1

z2 1 = 0
z 3 1


1
⇒ x n − n = 2Sinn α
x

⇒ xn +

1
= 2Cosnα
xn

33
...
a3 + b3 + c3 – 3abc = (a + b + c)
(a + bω + cω2) (a + bω2 + cω)

b
a

Sum of roots = − , product of roots
c
, discriminate = b2 – 4ac
a

20
...
If Z1, Z2, Z3 forms an equilateral triangle and Z0 is circum center
then
2

Z12 + Z 2 2 + Z32 = 3Z02 ,

22
...
Standard form of Quadratic equation is ax2 + bx +c = 0

3 2
Z
4
2

1
=Cosθ–iSinθ
x

Quadratic Expressions

18
...
Area of triangle formed by Z, ωZ, Z + ωZ
is

π
i
2

Cisβ
= Cis( α + β)
Cisβ

= −i,(1 + i) 2 = 2i,(1 − i) 2 = −2i

17
...
(Cosθ + iSinθ)n = Cosnθ + iSinnθ
31
...
Cisβ=Cis (α+ β),

1+ i
1− i
= i,
1− i
1+ i

+1

H
π

e 2 = i,log i =

12
...
Arg z = − Argz

n

z1 − z2 AB iθ
=
e
z1 − z3 AC

29
...
(1 + i) n + (1 − i)n = 2 2 Cos

AB, AC then

I

it is less, then it represents hyperbola
28
...
Arg z1z2 = Arg z1 + Argz2

=

kz 2 ± z1
k ±1

27
...
Arg of –x – iy is θ = −π + tan −1 for every
x
x > 0, y > 0

a + ib =

ant

If k = 1 the locus of z represents a line or perpendicular bisector
...
i = −1,

α 2 − β where α is nonreal complex and β is const

With radius

ends of diameter

y
for every
x

for every

25
...
Arg of –x + iy is θ = π − tan −1
x

24
...
If z − z1 = k (k≠1) represents circle with

b
6
...
Arg of x + iy is θ = tan −1 for every
x

8
...
Distance between two vertices
Z1, Z2 is
...
If the roots of ax2 + bx + c = 0 are
1, c a then a + b + c = 0
3
...
If one root of ax2 + bx + c = 0 is square of the other then ac2 + a2c
+ b3 = 3abc
5
...
If a1, a2,
...
sakshieducation
...
sakshieducation
...
If f(x) = (x + y)n then sum of coefficients is equal to f(1)
11
...
If a2 + b2 + c2 = K then range of

to

 −K 
ab + bc + ca is  , K 
 2


12
...
If the two roots are negative, then a, b, c will have same sign
9
...
f(x) = 0 is a polynomial then the equation whose roots are reciprocal of the roots of
 
f(x) = 0 is f   = 0 increased by 'K' is
x

f (1) + f (−1)
2
n
n
n
13
...
P (n–2r)2 =n + 2

to

14
...



2

1

12
...

14
...


f(x – K), multiplied by K is f(x/K)
For a, b, h ∈ R the roots of
(a – x) (b – x) = h2 are real and unequal
For a, b, c ∈ R the roots of
(x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are real and
unequal
Three roots of a cubical equation are A
...
P, a–3d, a–d, a+d, a+3d
If three roots are in G
...
Number of terms in the expansion
(x + a)n is n + 1
2
...
+ x r ) is n +r −1 Cr −1
3
...
In the expansion (x + y)n if n is odd greatn
n
est coefficients are C n2−1 , Cn2+1 if n is odd

18
...
In above, the term containing xs is

np − s
+1
p+q

6
...

7
...
(x+n)=n
8
...
(x+n)

n (n + 1)(n − 1)(3n + 2 )
24

of

(1+

x)n

General

notation

7
...
If 'A' is a given matrix, every square mat-rix can be expressed as a
sum of symme-tric and skew symmetric matrix where
Symmetric part = A + A

T

2

unsymmetric part =

n

b 

4
...
Coefficient of xn–2 in above is

expansion

C0 = Co , n C1 = C1 , n Cr = Cr

MATRICES
1
...
A square matrix is said to be a scalar matrix if all the elements in
the principal diagonal are equal and Other elements are zero's
3
...
A square matrix A is said to be Idem-potent matrix if A2 = A,
5
...
A square matrix A is said to be Symm-etric matrix if A = AT
A square matrix A is said to be Skew symmetric matrix if A=-AT

Tr +1 n − r + 1
=
Tr
r

n n + 1)
is (
2

Cn

S

−b
na

Binomial Theorem And Partial Fractions

n

n

greatest coefficient is

K

A

to eliminate second term roots are

In (x + a ) ,

term
...
= 2 n −1

2
(iii) α 4 + β4 + γ4 = s14 − 4s12 s2 + 4s1 s3 + 2s2

diminished by

2

16
...
Sum of odd binomial coefficients

(ii) α + β + γ = s − 2s2

(v) In ax n + bx n −1 + cxn −2
...
= 2 n −1

2
1

(iv) α 3 + β3 + γ3 = s31 − 3s1s2 + 3s3

th

20
...
For ax3 + bx2 + cx + d = 0
(i) Σα2β = (αβ + βγ + γα)
(α + β + γ) –3αβ γ = s1s2 – 3s3
2

n +1
2

Co + C1 + C2 +
...
Sum of binomial coefficients

16
...
P

2



15
...


f (1) − f (−1)
2

I

1
1
1
+
+
...
+ a n )

A + AT
2

9
...
A square matrix 'A' is said to be a singular matrix if det A = 0
11
...
If 'A' is a square matrix then det A=det AT
13
...

15
...


If AB = I = BA then A and B are called inverses of each other
(A-1)-1 = A, (AB)-1 = B-1A-1
If A and AT are invertible then (AT)-1 = (A-1)T
If A is non singular of order 3, A
is invertible, then A −1 =

17
...
(A-1)-1=A, (AB)-1=B-1 A-1, (AT)-1 =(A-1)T (ABC)-1 = C-1 B-1

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com

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com
A-1
...
If A and B are two non-singular matrices of the same type then
(i) Adj (AB) = (Adj B) (Adj A)
(ii) |Adj (AB) | = |Adj A| |Adj B |
= |Adj B| |Adj A|
20
...
C
 f 





where f = frequency of first quarfile class
F = cumulative frequency of the class just preceding to first quartile class
vi) upperQuartiledeviation

1 2 3 4 
3
y
k

l = lower limit of modal class with maximum frequency
f1 = frequency preceding modal class
f2 = frequency successive modal class
f3 = frequency of modal class
viii) Mode = 3Median - 2Mean

S

No of non zero rows=n= Rank of a matrix
If the system of equations AX=B is consistent if the coeff matrix A
and augmented matrix K are of same rank
Let AX = B be a system of equations of 'n' unknowns and ranks of
coeff matrix = r1 and rank of augmented matrix = r2
If r1≠ r2, then AX = B is inconsistant,
i
...
it has no solution
If r1= r2= n then AX=B is consistant, it has unique solution
If r1= r2 < n then AX=B is consistant and it has infinitely many
number of solutions

ix) Quartile deviation = Q3 − Q1

Q −Q

= Q3 + Q1
3

A

S

i) probability of occurrence = p
ii) probability of non occurrence = q
iii) p + q = 1
iv) probability of 'x' successes

1
...
an are said to be linearly independent
if are exists scalars x1 , x2
...

Such that x1 a1 + x2 a2 +
...
= xn = 0

2
...
an are said to be linearly dependent
if there

x , then its Arithmetic Mean x =

n

ii) For individual series If A is assumed
average then A
...


4
...


∑ xi

where (d i = xi − A )

VECTORS

x1 a1 + x2 a2 +
...
If number of trials are large and probab-ility of success is very
small then poisson distribution is used and given as

4
...
xn are n values of variant

Range

= Maximum + Minimum

P ( x = xi ) = nC x q n −x p x

e−λ λ k
k

1

xi) coefficient of Range

1
...
If n be positive integer p be a real number such that 0≤ P ≤ 1 a random variable X with range (0,1,2,----n) is said to follows binomial distribution
...
C
f







f m − f1 

...
e
...

7
...
n
And determinant = 0
Any two collinear vectors, any three coplanar vectors are linearly
dependent
...

Vector equation of sphere with center at c and radius a
2
2
2
2
is (r − c ) = a 2 or r − 2r
...

a, b are ends of diameter then equation of sphere (r − a )(r − b )= 0
If a, b are unit vectors then unit vector along bisector of ∠ AOB is
a+b
a+b

or

(a + b )

± a +b

8
...
If 'I' is in centre of ∆ ABC then,

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com

www
...
com

BC IA + CA IB + AB IC = 0

10
...
If 'S' is circum centre, 'O' is orthocenter of ∆ ABC
then, OA + OB + OC = 2OS
12
...
Let, a ≠ 0 b be two vectors
...
a
ii) The projection of b on a is b
...
a
...
b > 0 ⇒ 0 < θ < 90° ⇒ θ is acute

ii) a
...
b = 0 ⇒ θ = 90° ⇒ two vectors are ⊥ r to each other
...
In a right angled ∆ ABC, if AB is the hypotenuse and AB = P then
AB
...
CA + CA
...
∆ABC

is equilateral triangle of side 'a' then

AB
...
BC + BC
...
AB =

a

2

a
...
If a,b are not parallel then a×b is perpendicular to both of the vectors a,b
...
If a,b are not parallel then a
...

32
...
b ) and hence
33
...
If a,b are two vectors then a×b = - b×a
...
a×b = -b×a is called anticommutative law
...
If a,b are two nonzero vectors, then
sin (a, b ) =

a×b
a b

37
...
a )a

29
...
a
a
(b
...
i) The component of b on a is

1
(a × b ) and scalar area is 1 [a × b ]
2
2

16
...
i ) + (a
...
k ) = a ;
2

2

2

2

(a × i ) + (a × j ) + (a × k ) = 2 a
2

2

2

38
...
Vector equation
...

s,t∈R and also given as

S
is  AB AC AP  = 0



c
i
...
Vector equation
...
If ABCD is a parallelogram AB = a, BC = b and then the vector area
of ABCD is la×bl
40
...
The perpendicular distance from a point P to the line joining the
AP × AB
AB

21
...
of a plane passing through three non-collinear
Points
...
Vector equation
...
V
...
Vector equation of a line passing through A (a ), B (b ) is r =(1-t)a
+tb
r
19
...
of line passing through a & ⊥ to b, c

( ) and

B b

42
...

43
...
Volume of parallelopiped = [abc] with a, b, c as coterminus edges
...
The volume of the tetrahedron ABCD is
1
±  AB AC AD 

6

46
...
Vector equation of plane, at distance p (p >0) from origin and ⊥
to n is r
...
Perpendicular distance from origin to plane passing through a,b,c

1
6

tetrahedron = ± [a b c ]
47
...
The shortest distance between the skew

25
...
Vector equation of plane passing through A,B,C with position vectors a,b,c is [ r - a, b-a, c-a] =0 and r
...
If i,j,k are unit vectors then [i j k] = 1
50
...
sakshieducation
...
sakshieducation
...
[a×b, b×c, c×a] = (abc)2
52
...


(

)(

)

a×b
...


a
...
d
b
...
d

55
...
a = abc , b = abc , c = abc
[ ]
[ ]
[ ]

are called reciprocal system of vectors
57
...
Three vectors are coplanar if det = 0
If ai + j + k, i + bj + k, i + j + ck where a ≠ b ≠ c ≠ 1 are coplanar
then
1

+

+

1

1

+

1

+

1

=2

ii) ab bc ca

=1

I

1

i) 1 − a 1 − b 1 − c

Trigonometry:
In trigonometry, students usually find it diffi-cult to memorize the vast
number of formul-ae
...
The mo-re you practice, the more
ingrained in your br-ain these formulae will be, enabling you to re-call
them in any situation
...

Coordinate Geometry:
This section is usually considered easier than trigonometry
...
Pay att-ention to Locus and related topics, as the understanding of these makes coordinate Geome-try easy
...
Functions are the backbone of this section
...

Approximating sketches and graphical interp-retations will help you
solve problems faster
...

Algebra:
Don't use formulae to solve problems in topi-cs which are logic-oriented, such as permuta-tions and combinations, probability, location of
roots of a quadratic, geometrical applicati-ons of complex numbers,
vectors, and 3D-geometry
...
Try to do first and sec-ond level of
calculations mentally
You are going to appear for AIEEE this year, you must be very confident, don't pa-nic,it is not difficult and tough
...

Don't try to take up new topics as they con-sume time, you will also
lose your confide-nce on the topics that you have already pre-pared
...
If
you are confident about 60% of the questions, that will be enough to
get a good rank
...
Be wise, preplanning is very important
...

First try to finish all the simple questions to boost your Conf-idence
...
As you prepare for the board examinat-ion, you
should also prepare and solve the last year question papers for
AIEEE
...
You will gradually become
confident
...

Most of the questions in AIEEE are not dif-ficult
...

Each question has an element of sur-prise in it & a student who is
adept in tack-ling 'surprise questions' is most likely to sail through
successfully
...
There is a limit to which you can improve your
speed and strike rate beyond which what becomes very important is
your selec-tion of question
...
To optimize your performance you should quickly scan for easy questions and come back

H

2

53
...
b = a b

to the difficult ones later
...
So if you fo-cus on easy and average question i
...
85% of the
questions, you can easily score 70% marks without even attempting
difficult qu-estions
...


AIEEE 2009 Mathematics Section Analysis of CBSE syllabus
Of all the three sections in the AIEEE 2009 paper, the Mathematics
section was the toughest
...
Many candidates struggled with the Calculus and Coordinate Geometry portions
...
of Questions
Trigonometry
1
Algebra (XI)
6
Coordinate Geometry
5
Statistics
3
3-D (XI)
1
Class XII Syllabus
Topic
No
...
sakshieducation

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